Welcome!

But if you take a Human being, for instance Platon, and you just cut his leg off and replace it with a synthetic prosthesis. Would this person still be Platon ?

This is a known philosophical discussion. See:
https://en.wikipedia.org/wiki/Ship_of_Theseus

I'd like to ask the question that I care more about - why does it matter? How can we apply the answer (either yes it's the same person; no it's not; or a different answer) to RealLife (tm) and become more effective.

There is an unclear split between epistemic rationality (the theoretical) and instrumental rationality (the practical). I think this falls on the epistemic side of "knowing the truth". I prefer my rationality on the instrumental side of things. although I felt I had to learn a lot of epistemic ideas first.

To be clear: why does it matter? and how can we apply this knowledge? are the questions worth caring about, shortly after the "wow this weird thing happened". Good luck on moving forward to these questions.